Apparatus and method for implementing flexible finite differences in a graphics processor

ABSTRACT

An apparatus and method are described for implementing flexible finite differences in a graphics processor. For example, one embodiment of a graphics processor comprises: pixel shading logic to perform pixel shading operations on pixels associated with a rasterized primitive using covered pixels and uncovered pixels; and helper pixel selection logic to select helper pixels in the rasterized primitive, the helper pixels to be used by the pixel sharing logic for gradient computations, wherein for one or more of the covered pixels, the helper pixel selection logic attempts to identify one or more suitable covered helper pixels and, if no suitable covered helper pixels exist, identifies one or more uncovered helper pixels.

BACKGROUND

Field of the Invention

This invention relates generally to the field of computer processors. More particularly, the invention relates to an apparatus and method for implementing flexible finite differences in a graphics processor unit (GPU).

Description of the Related Art

Pixel shaders (sometimes referred to as “fragment shaders”), compute color and other attributes of each pixel in a graphics processor unit (GPU). GPUs typically run pixel shading on 2×2 pixel quads, packing multiple such quads into one single instruction multiple data (SIMD) batch. It often happens that one or more pixels in a quad lie outside the rasterized primitive. Such pixels are referred to as “helper” pixels. The primary need for helper pixels is for gradient computation by finite differencing, i.e., taking the difference between nearby pixels horizontally and vertically to estimate the rate at which a variable changes.

Gradients are required for the level of detail (LOD) computation in texture lookups, and also for certain shared instructions (e.g., such as ddx/ddy). However, helper pixels contribute in no other way to the visible image. Since shading helper pixels consumes valuable resources (power, bandwidth, and compute), it is desirable to find less expensive ways to compute gradients. The number of helper pixels in quad-based pixel shading can be substantial, often in the range of 40-50% or more of the total shaded pixels. Additionally, the problem of helper pixels is amplified for smaller triangles, and/or with the use of larger shading quads (i.e. coarse pixel shading (CPS)).

BRIEF DESCRIPTION OF THE DRAWINGS

A better understanding of the present invention can be obtained from the following detailed description in conjunction with the following drawings, in which:

FIG. 1 is a block diagram of an embodiment of a computer system with a processor having one or more processor cores and graphics processors;

FIG. 2 is a block diagram of one embodiment of a processor having one or more processor cores, an integrated memory controller, and an integrated graphics processor;

FIG. 3 is a block diagram of one embodiment of a graphics processor which may be a discreet graphics processing unit, or may be graphics processor integrated with a plurality of processing cores;

FIG. 4 is a block diagram of an embodiment of a graphics-processing engine for a graphics processor;

FIG. 5 is a block diagram of another embodiment of a graphics processor;

FIG. 6 is a block diagram of thread execution logic including an array of processing elements;

FIG. 7 illustrates a graphics processor execution unit instruction format according to an embodiment;

FIG. 8 is a block diagram of another embodiment of a graphics processor which includes a graphics pipeline, a media pipeline, a display engine, thread execution logic, and a render output pipeline;

FIG. 9A is a block diagram illustrating a graphics processor command format according to an embodiment;

FIG. 9B is a block diagram illustrating a graphics processor command sequence according to an embodiment;

FIG. 10 illustrates exemplary graphics software architecture for a data processing system according to an embodiment;

FIG. 11 illustrates an exemplary IP core development system that may be used to manufacture an integrated circuit to perform operations according to an embodiment;

FIG. 12 illustrates an exemplary system on a chip integrated circuit that may be fabricated using one or more IP cores, according to an embodiment;

FIG. 13 illustrates exemplary covered pixels and helper pixels within a rasterizer stamp;

FIG. 14 illustrates one embodiment of a pixel shader with helper pixel selection logic;

FIGS. 15A-B illustrate exemplary horizontal difference and vertical difference helper pixel selections;

FIG. 16 illustrates an exemplary rasterizer stamp in which a significant number of uncovered pixels are selected as helper pixels;

FIG. 17 illustrates a rasterizer stamp showing a significant number of covered pixels selected as helper pixels in accordance with one embodiment of the invention;

FIG. 18 illustrates a method in accordance with one embodiment of the invention;

FIG. 19A illustrates another exemplary rasterizer stamp in which a significant number of uncovered pixels are selected as helper pixels;

FIG. 19B illustrates another exemplary rasterizer stamp showing a significant number of covered pixels selected as helper pixels in accordance with one embodiment of the invention;

FIG. 20 illustrates an exemplary rasterizer stamp in which some uncovered pixels and some covered pixels are selected as helper pixels in accordance with one embodiment of the invention; and

FIG. 21 illustrates reductions in shading operations realized by the embodiments of the invention.

DETAILED DESCRIPTION

In the following description, for the purposes of explanation, numerous specific details are set forth in order to provide a thorough understanding of the embodiments of the invention described below. It will be apparent, however, to one skilled in the art that the embodiments of the invention may be practiced without some of these specific details. In other instances, well-known structures and devices are shown in block diagram form to avoid obscuring the underlying principles of the embodiments of the invention.

Exemplary Graphics Processor Architectures and Data Types

System Overview

FIG. 1 is a block diagram of a processing system 100, according to an embodiment. In various embodiments the system 100 includes one or more processors 102 and one or more graphics processors 108, and may be a single processor desktop system, a multiprocessor workstation system, or a server system having a large number of processors 102 or processor cores 107. In on embodiment, the system 100 is a processing platform incorporated within a system-on-a-chip (SoC) integrated circuit for use in mobile, handheld, or embedded devices.

An embodiment of system 100 can include, or be incorporated within a server-based gaming platform, a game console, including a game and media console, a mobile gaming console, a handheld game console, or an online game console. In some embodiments system 100 is a mobile phone, smart phone, tablet computing device or mobile Internet device. Data processing system 100 can also include, couple with, or be integrated within a wearable device, such as a smart watch wearable device, smart eyewear device, augmented reality device, or virtual reality device. In some embodiments, data processing system 100 is a television or set top box device having one or more processors 102 and a graphical interface generated by one or more graphics processors 108.

In some embodiments, the one or more processors 102 each include one or more processor cores 107 to process instructions which, when executed, perform operations for system and user software. In some embodiments, each of the one or more processor cores 107 is configured to process a specific instruction set 109. In some embodiments, instruction set 109 may facilitate Complex Instruction Set Computing (CISC), Reduced Instruction Set Computing (RISC), or computing via a Very Long Instruction Word (VLIW). Multiple processor cores 107 may each process a different instruction set 109, which may include instructions to facilitate the emulation of other instruction sets. Processor core 107 may also include other processing devices, such a Digital Signal Processor (DSP).

In some embodiments, the processor 102 includes cache memory 104. Depending on the architecture, the processor 102 can have a single internal cache or multiple levels of internal cache. In some embodiments, the cache memory is shared among various components of the processor 102. In some embodiments, the processor 102 also uses an external cache (e.g., a Level-3 (L3) cache or Last Level Cache (LLC)) (not shown), which may be shared among processor cores 107 using known cache coherency techniques. A register file 106 is additionally included in processor 102 which may include different types of registers for storing different types of data (e.g., integer registers, floating point registers, status registers, and an instruction pointer register). Some registers may be general-purpose registers, while other registers may be specific to the design of the processor 102.

In some embodiments, processor 102 is coupled to a processor bus 110 to transmit communication signals such as address, data, or control signals between processor 102 and other components in system 100. In one embodiment the system 100 uses an exemplary ‘hub’ system architecture, including a memory controller hub 116 and an Input Output (I/O) controller hub 130. A memory controller hub 116 facilitates communication between a memory device and other components of system 100, while an I/O Controller Hub (ICH) 130 provides connections to I/O devices via a local I/O bus. In one embodiment, the logic of the memory controller hub 116 is integrated within the processor.

Memory device 120 can be a dynamic random access memory (DRAM) device, a static random access memory (SRAM) device, flash memory device, phase-change memory device, or some other memory device having suitable performance to serve as process memory. In one embodiment the memory device 120 can operate as system memory for the system 100, to store data 122 and instructions 121 for use when the one or more processors 102 executes an application or process. Memory controller hub 116 also couples with an optional external graphics processor 112, which may communicate with the one or more graphics processors 108 in processors 102 to perform graphics and media operations.

In some embodiments, ICH 130 enables peripherals to connect to memory device 120 and processor 102 via a high-speed I/O bus. The I/O peripherals include, but are not limited to, an audio controller 146, a firmware interface 128, a wireless transceiver 126 (e.g., Wi-Fi, Bluetooth), a data storage device 124 (e.g., hard disk drive, flash memory, etc.), and a legacy I/O controller 140 for coupling legacy (e.g., Personal System 2 (PS/2)) devices to the system. One or more Universal Serial Bus (USB) controllers 142 connect input devices, such as keyboard and mouse 144 combinations. A network controller 134 may also couple to ICH 130. In some embodiments, a high-performance network controller (not shown) couples to processor bus 110. It will be appreciated that the system 100 shown is exemplary and not limiting, as other types of data processing systems that are differently configured may also be used. For example, the I/O controller hub 130 may be integrated within the one or more processor 102, or the memory controller hub 116 and I/O controller hub 130 may be integrated into a discreet external graphics processor, such as the external graphics processor 112.

FIG. 2 is a block diagram of an embodiment of a processor 200 having one or more processor cores 202A-202N, an integrated memory controller 214, and an integrated graphics processor 208. Those elements of FIG. 2 having the same reference numbers (or names) as the elements of any other figure herein can operate or function in any manner similar to that described elsewhere herein, but are not limited to such. Processor 200 can include additional cores up to and including additional core 202N represented by the dashed lined boxes. Each of processor cores 202A-202N includes one or more internal cache units 204A-204N. In some embodiments each processor core also has access to one or more shared cached units 206.

The internal cache units 204A-204N and shared cache units 206 represent a cache memory hierarchy within the processor 200. The cache memory hierarchy may include at least one level of instruction and data cache within each processor core and one or more levels of shared mid-level cache, such as a Level 2 (L2), Level 3 (L3), Level 4 (L4), or other levels of cache, where the highest level of cache before external memory is classified as the LLC. In some embodiments, cache coherency logic maintains coherency between the various cache units 206 and 204A-204N.

In some embodiments, processor 200 may also include a set of one or more bus controller units 216 and a system agent core 210. The one or more bus controller units 216 manage a set of peripheral buses, such as one or more Peripheral Component Interconnect buses (e.g., PCI, PCI Express). System agent core 210 provides management functionality for the various processor components. In some embodiments, system agent core 210 includes one or more integrated memory controllers 214 to manage access to various external memory devices (not shown).

In some embodiments, one or more of the processor cores 202A-202N include support for simultaneous multi-threading. In such embodiment, the system agent core 210 includes components for coordinating and operating cores 202A-202N during multi-threaded processing. System agent core 210 may additionally include a power control unit (PCU), which includes logic and components to regulate the power state of processor cores 202A-202N and graphics processor 208.

In some embodiments, processor 200 additionally includes graphics processor 208 to execute graphics processing operations. In some embodiments, the graphics processor 208 couples with the set of shared cache units 206, and the system agent core 210, including the one or more integrated memory controllers 214. In some embodiments, a display controller 211 is coupled with the graphics processor 208 to drive graphics processor output to one or more coupled displays. In some embodiments, display controller 211 may be a separate module coupled with the graphics processor via at least one interconnect, or may be integrated within the graphics processor 208 or system agent core 210.

In some embodiments, a ring based interconnect unit 212 is used to couple the internal components of the processor 200. However, an alternative interconnect unit may be used, such as a point-to-point interconnect, a switched interconnect, or other techniques, including techniques well known in the art. In some embodiments, graphics processor 208 couples with the ring interconnect 212 via an I/O link 213.

The exemplary I/O link 213 represents at least one of multiple varieties of I/O interconnects, including an on package I/O interconnect which facilitates communication between various processor components and a high-performance embedded memory module 218, such as an eDRAM module. In some embodiments, each of the processor cores 202-202N and graphics processor 208 use embedded memory modules 218 as a shared Last Level Cache.

In some embodiments, processor cores 202A-202N are homogenous cores executing the same instruction set architecture. In another embodiment, processor cores 202A-202N are heterogeneous in terms of instruction set architecture (ISA), where one or more of processor cores 202A-N execute a first instruction set, while at least one of the other cores executes a subset of the first instruction set or a different instruction set. In one embodiment processor cores 202A-202N are heterogeneous in terms of microarchitecture, where one or more cores having a relatively higher power consumption couple with one or more power cores having a lower power consumption. Additionally, processor 200 can be implemented on one or more chips or as an SoC integrated circuit having the illustrated components, in addition to other components.

FIG. 3 is a block diagram of a graphics processor 300, which may be a discrete graphics processing unit, or may be a graphics processor integrated with a plurality of processing cores. In some embodiments, the graphics processor communicates via a memory mapped I/O interface to registers on the graphics processor and with commands placed into the processor memory. In some embodiments, graphics processor 300 includes a memory interface 314 to access memory. Memory interface 314 can be an interface to local memory, one or more internal caches, one or more shared external caches, and/or to system memory.

In some embodiments, graphics processor 300 also includes a display controller 302 to drive display output data to a display device 320. Display controller 302 includes hardware for one or more overlay planes for the display and composition of multiple layers of video or user interface elements. In some embodiments, graphics processor 300 includes a video codec engine 306 to encode, decode, or transcode media to, from, or between one or more media encoding formats, including, but not limited to Moving Picture Experts Group (MPEG) formats such as MPEG-2, Advanced Video Coding (AVC) formats such as H.264/MPEG-4 AVC, as well as the Society of Motion Picture & Television Engineers (SMPTE) 421 M/VC-1, and Joint Photographic Experts Group (JPEG) formats such as JPEG, and Motion JPEG (MJPEG) formats.

In some embodiments, graphics processor 300 includes a block image transfer (BLIT) engine 304 to perform two-dimensional (2D) rasterizer operations including, for example, bit-boundary block transfers. However, in one embodiment, 2D graphics operations are performed using one or more components of graphics processing engine (GPE) 310. In some embodiments, graphics processing engine 310 is a compute engine for performing graphics operations, including three-dimensional (3D) graphics operations and media operations.

In some embodiments, GPE 310 includes a 3D pipeline 312 for performing 3D operations, such as rendering three-dimensional images and scenes using processing functions that act upon 3D primitive shapes (e.g., rectangle, triangle, etc.). The 3D pipeline 312 includes programmable and fixed function elements that perform various tasks within the element and/or spawn execution threads to a 3D/Media sub-system 315. While 3D pipeline 312 can be used to perform media operations, an embodiment of GPE 310 also includes a media pipeline 316 that is specifically used to perform media operations, such as video post-processing and image enhancement.

In some embodiments, media pipeline 316 includes fixed function or programmable logic units to perform one or more specialized media operations, such as video decode acceleration, video de-interlacing, and video encode acceleration in place of, or on behalf of video codec engine 306. In some embodiments, media pipeline 316 additionally includes a thread spawning unit to spawn threads for execution on 3D/Media sub-system 315. The spawned threads perform computations for the media operations on one or more graphics execution units included in 3D/Media sub-system 315.

In some embodiments, 3D/Media subsystem 315 includes logic for executing threads spawned by 3D pipeline 312 and media pipeline 316. In one embodiment, the pipelines send thread execution requests to 3D/Media subsystem 315, which includes thread dispatch logic for arbitrating and dispatching the various requests to available thread execution resources. The execution resources include an array of graphics execution units to process the 3D and media threads. In some embodiments, 3D/Media subsystem 315 includes one or more internal caches for thread instructions and data. In some embodiments, the subsystem also includes shared memory, including registers and addressable memory, to share data between threads and to store output data.

3D/Media Processing

FIG. 4 is a block diagram of a graphics processing engine 410 of a graphics processor in accordance with some embodiments. In one embodiment, the GPE 410 is a version of the GPE 310 shown in FIG. 3. Elements of FIG. 4 having the same reference numbers (or names) as the elements of any other figure herein can operate or function in any manner similar to that described elsewhere herein, but are not limited to such.

In some embodiments, GPE 410 couples with a command streamer 403, which provides a command stream to the GPE 3D and media pipelines 412, 416. In some embodiments, command streamer 403 is coupled to memory, which can be system memory, or one or more of internal cache memory and shared cache memory. In some embodiments, command streamer 403 receives commands from the memory and sends the commands to 3D pipeline 412 and/or media pipeline 416. The commands are directives fetched from a ring buffer, which stores commands for the 3D and media pipelines 412, 416. In one embodiment, the ring buffer can additionally include batch command buffers storing batches of multiple commands. The 3D and media pipelines 412, 416 process the commands by performing operations via logic within the respective pipelines or by dispatching one or more execution threads to an execution unit array 414. In some embodiments, execution unit array 414 is scalable, such that the array includes a variable number of execution units based on the target power and performance level of GPE 410.

In some embodiments, a sampling engine 430 couples with memory (e.g., cache memory or system memory) and execution unit array 414. In some embodiments, sampling engine 430 provides a memory access mechanism for execution unit array 414 that allows execution array 414 to read graphics and media data from memory. In some embodiments, sampling engine 430 includes logic to perform specialized image sampling operations for media.

In some embodiments, the specialized media sampling logic in sampling engine 430 includes a de-noise/de-interlace module 432, a motion estimation module 434, and an image scaling and filtering module 436. In some embodiments, de-noise/de-interlace module 432 includes logic to perform one or more of a de-noise or a de-interlace algorithm on decoded video data. The de-interlace logic combines alternating fields of interlaced video content into a single fame of video. The de-noise logic reduces or removes data noise from video and image data. In some embodiments, the de-noise logic and de-interlace logic are motion adaptive and use spatial or temporal filtering based on the amount of motion detected in the video data. In some embodiments, the de-noise/de-interlace module 432 includes dedicated motion detection logic (e.g., within the motion estimation engine 434).

In some embodiments, motion estimation engine 434 provides hardware acceleration for video operations by performing video acceleration functions such as motion vector estimation and prediction on video data. The motion estimation engine determines motion vectors that describe the transformation of image data between successive video frames. In some embodiments, a graphics processor media codec uses video motion estimation engine 434 to perform operations on video at the macro-block level that may otherwise be too computationally intensive to perform with a general-purpose processor. In some embodiments, motion estimation engine 434 is generally available to graphics processor components to assist with video decode and processing functions that are sensitive or adaptive to the direction or magnitude of the motion within video data.

In some embodiments, image scaling and filtering module 436 performs image-processing operations to enhance the visual quality of generated images and video. In some embodiments, scaling and filtering module 436 processes image and video data during the sampling operation before providing the data to execution unit array 414.

In some embodiments, the GPE 410 includes a data port 444, which provides an additional mechanism for graphics subsystems to access memory. In some embodiments, data port 444 facilitates memory access for operations including render target writes, constant buffer reads, scratch memory space reads/writes, and media surface accesses. In some embodiments, data port 444 includes cache memory space to cache accesses to memory. The cache memory can be a single data cache or separated into multiple caches for the multiple subsystems that access memory via the data port (e.g., a render buffer cache, a constant buffer cache, etc.). In some embodiments, threads executing on an execution unit in execution unit array 414 communicate with the data port by exchanging messages via a data distribution interconnect that couples each of the sub-systems of GPE 410.

Execution Units

FIG. 5 is a block diagram of another embodiment of a graphics processor 500. Elements of FIG. 5 having the same reference numbers (or names) as the elements of any other figure herein can operate or function in any manner similar to that described elsewhere herein, but are not limited to such.

In some embodiments, graphics processor 500 includes a ring interconnect 502, a pipeline front-end 504, a media engine 537, and graphics cores 580A-580N. In some embodiments, ring interconnect 502 couples the graphics processor to other processing units, including other graphics processors or one or more general-purpose processor cores. In some embodiments, the graphics processor is one of many processors integrated within a multi-core processing system.

In some embodiments, graphics processor 500 receives batches of commands via ring interconnect 502. The incoming commands are interpreted by a command streamer 503 in the pipeline front-end 504. In some embodiments, graphics processor 500 includes scalable execution logic to perform 3D geometry processing and media processing via the graphics core(s) 580A-580N. For 3D geometry processing commands, command streamer 503 supplies commands to geometry pipeline 536. For at least some media processing commands, command streamer 503 supplies the commands to a video front end 534, which couples with a media engine 537. In some embodiments, media engine 537 includes a Video Quality Engine (VQE) 530 for video and image post-processing and a multi-format encode/decode (MFX) 533 engine to provide hardware-accelerated media data encode and decode. In some embodiments, geometry pipeline 536 and media engine 537 each generate execution threads for the thread execution resources provided by at least one graphics core 580A.

In some embodiments, graphics processor 500 includes scalable thread execution resources featuring modular cores 580A-580N (sometimes referred to as core slices), each having multiple sub-cores 550A-550N, 560A-560N (sometimes referred to as core sub-slices). In some embodiments, graphics processor 500 can have any number of graphics cores 580A through 580N. In some embodiments, graphics processor 500 includes a graphics core 580A having at least a first sub-core 550A and a second core sub-core 560A. In other embodiments, the graphics processor is a low power processor with a single sub-core (e.g., 550A). In some embodiments, graphics processor 500 includes multiple graphics cores 580A-580N, each including a set of first sub-cores 550A-550N and a set of second sub-cores 560A-560N. Each sub-core in the set of first sub-cores 550A-550N includes at least a first set of execution units 552A-552N and media/texture samplers 554A-554N. Each sub-core in the set of second sub-cores 560A-560N includes at least a second set of execution units 562A-562N and samplers 564A-564N. In some embodiments, each sub-core 550A-550N, 560A-560N shares a set of shared resources 570A-570N. In some embodiments, the shared resources include shared cache memory and pixel operation logic. Other shared resources may also be included in the various embodiments of the graphics processor.

FIG. 6 illustrates thread execution logic 600 including an array of processing elements employed in some embodiments of a GPE. Elements of FIG. 6 having the same reference numbers (or names) as the elements of any other figure herein can operate or function in any manner similar to that described elsewhere herein, but are not limited to such.

In some embodiments, thread execution logic 600 includes a pixel shader 602, a thread dispatcher 604, instruction cache 606, a scalable execution unit array including a plurality of execution units 608A-608N, a sampler 610, a data cache 612, and a data port 614. In one embodiment the included components are interconnected via an interconnect fabric that links to each of the components. In some embodiments, thread execution logic 600 includes one or more connections to memory, such as system memory or cache memory, through one or more of instruction cache 606, data port 614, sampler 610, and execution unit array 608A-608N. In some embodiments, each execution unit (e.g. 608A) is an individual vector processor capable of executing multiple simultaneous threads and processing multiple data elements in parallel for each thread. In some embodiments, execution unit array 608A-608N includes any number individual execution units.

In some embodiments, execution unit array 608A-608N is primarily used to execute “shader” programs. In some embodiments, the execution units in array 608A-608N execute an instruction set that includes native support for many standard 3D graphics shader instructions, such that shader programs from graphics libraries (e.g., Direct 3D and OpenGL) are executed with a minimal translation. The execution units support vertex and geometry processing (e.g., vertex programs, geometry programs, vertex shaders), pixel processing (e.g., pixel shaders, fragment shaders) and general-purpose processing (e.g., compute and media shaders).

Each execution unit in execution unit array 608A-608N operates on arrays of data elements. The number of data elements is the “execution size,” or the number of channels for the instruction. An execution channel is a logical unit of execution for data element access, masking, and flow control within instructions. The number of channels may be independent of the number of physical Arithmetic Logic Units (ALUs) or Floating Point Units (FPUs) for a particular graphics processor. In some embodiments, execution units 608A-608N support integer and floating-point data types.

The execution unit instruction set includes single instruction multiple data (SIMD) instructions. The various data elements can be stored as a packed data type in a register and the execution unit will process the various elements based on the data size of the elements. For example, when operating on a 256-bit wide vector, the 256 bits of the vector are stored in a register and the execution unit operates on the vector as four separate 64-bit packed data elements (Quad-Word (QW) size data elements), eight separate 32-bit packed data elements (Double Word (DW) size data elements), sixteen separate 16-bit packed data elements (Word (W) size data elements), or thirty-two separate 8-bit data elements (byte (B) size data elements). However, different vector widths and register sizes are possible.

One or more internal instruction caches (e.g., 606) are included in the thread execution logic 600 to cache thread instructions for the execution units. In some embodiments, one or more data caches (e.g., 612) are included to cache thread data during thread execution. In some embodiments, sampler 610 is included to provide texture sampling for 3D operations and media sampling for media operations. In some embodiments, sampler 610 includes specialized texture or media sampling functionality to process texture or media data during the sampling process before providing the sampled data to an execution unit.

During execution, the graphics and media pipelines send thread initiation requests to thread execution logic 600 via thread spawning and dispatch logic. In some embodiments, thread execution logic 600 includes a local thread dispatcher 604 that arbitrates thread initiation requests from the graphics and media pipelines and instantiates the requested threads on one or more execution units 608A-608N. For example, the geometry pipeline (e.g., 536 of FIG. 5) dispatches vertex processing, tessellation, or geometry processing threads to thread execution logic 600 (FIG. 6). In some embodiments, thread dispatcher 604 can also process runtime thread spawning requests from the executing shader programs.

Once a group of geometric objects has been processed and rasterized into pixel data, pixel shader 602 is invoked to further compute output information and cause results to be written to output surfaces (e.g., color buffers, depth buffers, stencil buffers, etc.). In some embodiments, pixel shader 602 calculates the values of the various vertex attributes that are to be interpolated across the rasterized object. In some embodiments, pixel shader 602 then executes an application programming interface (API)-supplied pixel shader program. To execute the pixel shader program, pixel shader 602 dispatches threads to an execution unit (e.g., 608A) via thread dispatcher 604. In some embodiments, pixel shader 602 uses texture sampling logic in sampler 610 to access texture data in texture maps stored in memory. Arithmetic operations on the texture data and the input geometry data compute pixel color data for each geometric fragment, or discards one or more pixels from further processing.

In some embodiments, the data port 614 provides a memory access mechanism for the thread execution logic 600 output processed data to memory for processing on a graphics processor output pipeline. In some embodiments, the data port 614 includes or couples to one or more cache memories (e.g., data cache 612) to cache data for memory access via the data port.

FIG. 7 is a block diagram illustrating a graphics processor instruction formats 700 according to some embodiments. In one or more embodiment, the graphics processor execution units support an instruction set having instructions in multiple formats. The solid lined boxes illustrate the components that are generally included in an execution unit instruction, while the dashed lines include components that are optional or that are only included in a sub-set of the instructions. In some embodiments, instruction format 700 described and illustrated are macro-instructions, in that they are instructions supplied to the execution unit, as opposed to micro-operations resulting from instruction decode once the instruction is processed.

In some embodiments, the graphics processor execution units natively support instructions in a 128-bit format 710. A 64-bit compacted instruction format 730 is available for some instructions based on the selected instruction, instruction options, and number of operands. The native 128-bit format 710 provides access to all instruction options, while some options and operations are restricted in the 64-bit format 730. The native instructions available in the 64-bit format 730 vary by embodiment. In some embodiments, the instruction is compacted in part using a set of index values in an index field 713. The execution unit hardware references a set of compaction tables based on the index values and uses the compaction table outputs to reconstruct a native instruction in the 128-bit format 710.

For each format, instruction opcode 712 defines the operation that the execution unit is to perform. The execution units execute each instruction in parallel across the multiple data elements of each operand. For example, in response to an add instruction the execution unit performs a simultaneous add operation across each color channel representing a texture element or picture element. By default, the execution unit performs each instruction across all data channels of the operands. In some embodiments, instruction control field 714 enables control over certain execution options, such as channels selection (e.g., predication) and data channel order (e.g., swizzle). For 128-bit instructions 710 an exec-size field 716 limits the number of data channels that will be executed in parallel. In some embodiments, exec-size field 716 is not available for use in the 64-bit compact instruction format 730.

Some execution unit instructions have up to three operands including two source operands, src0 722, src1 722, and one destination 718. In some embodiments, the execution units support dual destination instructions, where one of the destinations is implied. Data manipulation instructions can have a third source operand (e.g., SRC2 724), where the instruction opcode 712 determines the number of source operands. An instruction's last source operand can be an immediate (e.g., hard-coded) value passed with the instruction.

In some embodiments, the 128-bit instruction format 710 includes an access/address mode information 726 specifying, for example, whether direct register addressing mode or indirect register addressing mode is used. When direct register addressing mode is used, the register address of one or more operands is directly provided by bits in the instruction 710.

In some embodiments, the 128-bit instruction format 710 includes an access/address mode field 726, which specifies an address mode and/or an access mode for the instruction. In one embodiment the access mode to define a data access alignment for the instruction. Some embodiments support access modes including a 16-byte aligned access mode and a 1-byte aligned access mode, where the byte alignment of the access mode determines the access alignment of the instruction operands. For example, when in a first mode, the instruction 710 may use byte-aligned addressing for source and destination operands and when in a second mode, the instruction 710 may use 16-byte-aligned addressing for all source and destination operands.

In one embodiment, the address mode portion of the access/address mode field 726 determines whether the instruction is to use direct or indirect addressing. When direct register addressing mode is used bits in the instruction 710 directly provide the register address of one or more operands. When indirect register addressing mode is used, the register address of one or more operands may be computed based on an address register value and an address immediate field in the instruction.

In some embodiments instructions are grouped based on opcode 712 bit-fields to simplify Opcode decode 740. For an 8-bit opcode, bits 4, 5, and 6 allow the execution unit to determine the type of opcode. The precise opcode grouping shown is merely an example. In some embodiments, a move and logic opcode group 742 includes data movement and logic instructions (e.g., move (mov), compare (cmp)). In some embodiments, move and logic group 742 shares the five most significant bits (MSB), where move (mov) instructions are in the form of 0000xxxxb and logic instructions are in the form of 0001xxxxb. A flow control instruction group 744 (e.g., call, jump (jmp)) includes instructions in the form of 0010xxxxb (e.g., 0x20). A miscellaneous instruction group 746 includes a mix of instructions, including synchronization instructions (e.g., wait, send) in the form of 0011xxxxb (e.g., 0x30). A parallel math instruction group 748 includes component-wise arithmetic instructions (e.g., add, multiply (mul)) in the form of 0100xxxxb (e.g., 0x40). The parallel math group 748 performs the arithmetic operations in parallel across data channels. The vector math group 750 includes arithmetic instructions (e.g., dp4) in the form of 0101xxxxb (e.g., 0x50). The vector math group performs arithmetic such as dot product calculations on vector operands.

Graphics Pipeline

FIG. 8 is a block diagram of another embodiment of a graphics processor 800. Elements of FIG. 8 having the same reference numbers (or names) as the elements of any other figure herein can operate or function in any manner similar to that described elsewhere herein, but are not limited to such.

In some embodiments, graphics processor 800 includes a graphics pipeline 820, a media pipeline 830, a display engine 840, thread execution logic 850, and a render output pipeline 870. In some embodiments, graphics processor 800 is a graphics processor within a multi-core processing system that includes one or more general purpose processing cores. The graphics processor is controlled by register writes to one or more control registers (not shown) or via commands issued to graphics processor 800 via a ring interconnect 802. In some embodiments, ring interconnect 802 couples graphics processor 800 to other processing components, such as other graphics processors or general-purpose processors. Commands from ring interconnect 802 are interpreted by a command streamer 803, which supplies instructions to individual components of graphics pipeline 820 or media pipeline 830.

In some embodiments, command streamer 803 directs the operation of a vertex fetcher 805 that reads vertex data from memory and executes vertex-processing commands provided by command streamer 803. In some embodiments, vertex fetcher 805 provides vertex data to a vertex shader 807, which performs coordinate space transformation and lighting operations to each vertex. In some embodiments, vertex fetcher 805 and vertex shader 807 execute vertex-processing instructions by dispatching execution threads to execution units 852A, 852B via a thread dispatcher 831.

In some embodiments, execution units 852A, 852B are an array of vector processors having an instruction set for performing graphics and media operations. In some embodiments, execution units 852A, 852B have an attached L1 cache 851 that is specific for each array or shared between the arrays. The cache can be configured as a data cache, an instruction cache, or a single cache that is partitioned to contain data and instructions in different partitions.

In some embodiments, graphics pipeline 820 includes tessellation components to perform hardware-accelerated tessellation of 3D objects. In some embodiments, a programmable hull shader 811 configures the tessellation operations. A programmable domain shader 817 provides back-end evaluation of tessellation output. A tessellator 813 operates at the direction of hull shader 811 and contains special purpose logic to generate a set of detailed geometric objects based on a coarse geometric model that is provided as input to graphics pipeline 820. In some embodiments, if tessellation is not used, tessellation components 811, 813, 817 can be bypassed.

In some embodiments, complete geometric objects can be processed by a geometry shader 819 via one or more threads dispatched to execution units 852A, 852B, or can proceed directly to the clipper 829. In some embodiments, the geometry shader operates on entire geometric objects, rather than vertices or patches of vertices as in previous stages of the graphics pipeline. If the tessellation is disabled the geometry shader 819 receives input from the vertex shader 807. In some embodiments, geometry shader 819 is programmable by a geometry shader program to perform geometry tessellation if the tessellation units are disabled.

Before rasterization, a clipper 829 processes vertex data. The clipper 829 may be a fixed function clipper or a programmable clipper having clipping and geometry shader functions. In some embodiments, a rasterizer and depth test component 873 in the render output pipeline 870 dispatches pixel shaders to convert the geometric objects into their per pixel representations. In some embodiments, pixel shader logic is included in thread execution logic 850. In some embodiments, an application can bypass the rasterizer 873 and access un-rasterized vertex data via a stream out unit 823.

The graphics processor 800 has an interconnect bus, interconnect fabric, or some other interconnect mechanism that allows data and message passing amongst the major components of the processor. In some embodiments, execution units 852A, 852B and associated cache(s) 851, texture and media sampler 854, and texture/sampler cache 858 interconnect via a data port 856 to perform memory access and communicate with render output pipeline components of the processor. In some embodiments, sampler 854, caches 851, 858 and execution units 852A, 852B each have separate memory access paths.

In some embodiments, render output pipeline 870 contains a rasterizer and depth test component 873 that converts vertex-based objects into an associated pixel-based representation. In some embodiments, the rasterizer logic includes a windower/masker unit to perform fixed function triangle and line rasterization. An associated render cache 878 and depth cache 879 are also available in some embodiments. A pixel operations component 877 performs pixel-based operations on the data, though in some instances, pixel operations associated with 2D operations (e.g. bit block image transfers with blending) are performed by the 2D engine 841, or substituted at display time by the display controller 843 using overlay display planes. In some embodiments, a shared L3 cache 875 is available to all graphics components, allowing the sharing of data without the use of main system memory.

In some embodiments, graphics processor media pipeline 830 includes a media engine 837 and a video front end 834. In some embodiments, video front end 834 receives pipeline commands from the command streamer 803. In some embodiments, media pipeline 830 includes a separate command streamer. In some embodiments, video front-end 834 processes media commands before sending the command to the media engine 837. In some embodiments, media engine 337 includes thread spawning functionality to spawn threads for dispatch to thread execution logic 850 via thread dispatcher 831.

In some embodiments, graphics processor 800 includes a display engine 840. In some embodiments, display engine 840 is external to processor 800 and couples with the graphics processor via the ring interconnect 802, or some other interconnect bus or fabric. In some embodiments, display engine 840 includes a 2D engine 841 and a display controller 843. In some embodiments, display engine 840 contains special purpose logic capable of operating independently of the 3D pipeline. In some embodiments, display controller 843 couples with a display device (not shown), which may be a system integrated display device, as in a laptop computer, or an external display device attached via a display device connector.

In some embodiments, graphics pipeline 820 and media pipeline 830 are configurable to perform operations based on multiple graphics and media programming interfaces and are not specific to any one application programming interface (API). In some embodiments, driver software for the graphics processor translates API calls that are specific to a particular graphics or media library into commands that can be processed by the graphics processor. In some embodiments, support is provided for the Open Graphics Library (OpenGL) and Open Computing Language (OpenCL) from the Khronos Group, the Direct3D library from the Microsoft Corporation, or support may be provided to both OpenGL and D3D. Support may also be provided for the Open Source Computer Vision Library (OpenCV). A future API with a compatible 3D pipeline would also be supported if a mapping can be made from the pipeline of the future API to the pipeline of the graphics processor.

Graphics Pipeline Programming

FIG. 9A is a block diagram illustrating a graphics processor command format 900 according to some embodiments. FIG. 9B is a block diagram illustrating a graphics processor command sequence 910 according to an embodiment. The solid lined boxes in FIG. 9A illustrate the components that are generally included in a graphics command while the dashed lines include components that are optional or that are only included in a sub-set of the graphics commands. The exemplary graphics processor command format 900 of FIG. 9A includes data fields to identify a target client 902 of the command, a command operation code (opcode) 904, and the relevant data 906 for the command. A sub-opcode 905 and a command size 908 are also included in some commands.

In some embodiments, client 902 specifies the client unit of the graphics device that processes the command data. In some embodiments, a graphics processor command parser examines the client field of each command to condition the further processing of the command and route the command data to the appropriate client unit. In some embodiments, the graphics processor client units include a memory interface unit, a render unit, a 2D unit, a 3D unit, and a media unit. Each client unit has a corresponding processing pipeline that processes the commands. Once the command is received by the client unit, the client unit reads the opcode 904 and, if present, sub-opcode 905 to determine the operation to perform. The client unit performs the command using information in data field 906. For some commands an explicit command size 908 is expected to specify the size of the command. In some embodiments, the command parser automatically determines the size of at least some of the commands based on the command opcode. In some embodiments commands are aligned via multiples of a double word.

The flow diagram in FIG. 9B shows an exemplary graphics processor command sequence 910. In some embodiments, software or firmware of a data processing system that features an embodiment of a graphics processor uses a version of the command sequence shown to set up, execute, and terminate a set of graphics operations. A sample command sequence is shown and described for purposes of example only as embodiments are not limited to these specific commands or to this command sequence. Moreover, the commands may be issued as batch of commands in a command sequence, such that the graphics processor will process the sequence of commands in at least partially concurrence.

In some embodiments, the graphics processor command sequence 910 may begin with a pipeline flush command 912 to cause any active graphics pipeline to complete the currently pending commands for the pipeline. In some embodiments, the 3D pipeline 922 and the media pipeline 924 do not operate concurrently. The pipeline flush is performed to cause the active graphics pipeline to complete any pending commands. In response to a pipeline flush, the command parser for the graphics processor will pause command processing until the active drawing engines complete pending operations and the relevant read caches are invalidated. Optionally, any data in the render cache that is marked ‘dirty’ can be flushed to memory. In some embodiments, pipeline flush command 912 can be used for pipeline synchronization or before placing the graphics processor into a low power state.

In some embodiments, a pipeline select command 913 is used when a command sequence requires the graphics processor to explicitly switch between pipelines. In some embodiments, a pipeline select command 913 is required only once within an execution context before issuing pipeline commands unless the context is to issue commands for both pipelines. In some embodiments, a pipeline flush command is 912 is required immediately before a pipeline switch via the pipeline select command 913.

In some embodiments, a pipeline control command 914 configures a graphics pipeline for operation and is used to program the 3D pipeline 922 and the media pipeline 924. In some embodiments, pipeline control command 914 configures the pipeline state for the active pipeline. In one embodiment, the pipeline control command 914 is used for pipeline synchronization and to clear data from one or more cache memories within the active pipeline before processing a batch of commands.

In some embodiments, return buffer state commands 916 are used to configure a set of return buffers for the respective pipelines to write data. Some pipeline operations require the allocation, selection, or configuration of one or more return buffers into which the operations write intermediate data during processing. In some embodiments, the graphics processor also uses one or more return buffers to store output data and to perform cross thread communication. In some embodiments, the return buffer state 916 includes selecting the size and number of return buffers to use for a set of pipeline operations.

The remaining commands in the command sequence differ based on the active pipeline for operations. Based on a pipeline determination 920, the command sequence is tailored to the 3D pipeline 922 beginning with the 3D pipeline state 930, or the media pipeline 924 beginning at the media pipeline state 940.

The commands for the 3D pipeline state 930 include 3D state setting commands for vertex buffer state, vertex element state, constant color state, depth buffer state, and other state variables that are to be configured before 3D primitive commands are processed. The values of these commands are determined at least in part based the particular 3D API in use. In some embodiments, 3D pipeline state 930 commands are also able to selectively disable or bypass certain pipeline elements if those elements will not be used.

In some embodiments, 3D primitive 932 command is used to submit 3D primitives to be processed by the 3D pipeline. Commands and associated parameters that are passed to the graphics processor via the 3D primitive 932 command are forwarded to the vertex fetch function in the graphics pipeline. The vertex fetch function uses the 3D primitive 932 command data to generate vertex data structures. The vertex data structures are stored in one or more return buffers. In some embodiments, 3D primitive 932 command is used to perform vertex operations on 3D primitives via vertex shaders. To process vertex shaders, 3D pipeline 922 dispatches shader execution threads to graphics processor execution units.

In some embodiments, 3D pipeline 922 is triggered via an execute 934 command or event. In some embodiments, a register write triggers command execution. In some embodiments execution is triggered via a ‘go’ or ‘kick’ command in the command sequence. In one embodiment command execution is triggered using a pipeline synchronization command to flush the command sequence through the graphics pipeline. The 3D pipeline will perform geometry processing for the 3D primitives. Once operations are complete, the resulting geometric objects are rasterized and the pixel engine colors the resulting pixels. Additional commands to control pixel shading and pixel back end operations may also be included for those operations.

In some embodiments, the graphics processor command sequence 910 follows the media pipeline 924 path when performing media operations. In general, the specific use and manner of programming for the media pipeline 924 depends on the media or compute operations to be performed. Specific media decode operations may be offloaded to the media pipeline during media decode. In some embodiments, the media pipeline can also be bypassed and media decode can be performed in whole or in part using resources provided by one or more general purpose processing cores. In one embodiment, the media pipeline also includes elements for general-purpose graphics processor unit (GPGPU) operations, where the graphics processor is used to perform SIMD vector operations using computational shader programs that are not explicitly related to the rendering of graphics primitives.

In some embodiments, media pipeline 924 is configured in a similar manner as the 3D pipeline 922. A set of media pipeline state commands 940 are dispatched or placed into in a command queue before the media object commands 942. In some embodiments, media pipeline state commands 940 include data to configure the media pipeline elements that will be used to process the media objects. This includes data to configure the video decode and video encode logic within the media pipeline, such as encode or decode format. In some embodiments, media pipeline state commands 940 also support the use one or more pointers to “indirect” state elements that contain a batch of state settings.

In some embodiments, media object commands 942 supply pointers to media objects for processing by the media pipeline. The media objects include memory buffers containing video data to be processed. In some embodiments, all media pipeline states must be valid before issuing a media object command 942. Once the pipeline state is configured and media object commands 942 are queued, the media pipeline 924 is triggered via an execute command 944 or an equivalent execute event (e.g., register write). Output from media pipeline 924 may then be post processed by operations provided by the 3D pipeline 922 or the media pipeline 924. In some embodiments, GPGPU operations are configured and executed in a similar manner as media operations.

Graphics Software Architecture

FIG. 10 illustrates exemplary graphics software architecture for a data processing system 1000 according to some embodiments. In some embodiments, software architecture includes a 3D graphics application 1010, an operating system 1020, and at least one processor 1030. In some embodiments, processor 1030 includes a graphics processor 1032 and one or more general-purpose processor core(s) 1034. The graphics application 1010 and operating system 1020 each execute in the system memory 1050 of the data processing system.

In some embodiments, 3D graphics application 1010 contains one or more shader programs including shader instructions 1012. The shader language instructions may be in a high-level shader language, such as the High Level Shader Language (HLSL) or the OpenGL Shader Language (GLSL). The application also includes executable instructions 1014 in a machine language suitable for execution by the general-purpose processor core 1034. The application also includes graphics objects 1016 defined by vertex data.

In some embodiments, operating system 1020 is a Microsoft® Windows® operating system from the Microsoft Corporation, a proprietary UNIX-like operating system, or an open source UNIX-like operating system using a variant of the Linux kernel. When the Direct3D API is in use, the operating system 1020 uses a front-end shader compiler 1024 to compile any shader instructions 1012 in HLSL into a lower-level shader language. The compilation may be a just-in-time (JIT) compilation or the application can perform shader pre-compilation. In some embodiments, high-level shaders are compiled into low-level shaders during the compilation of the 3D graphics application 1010.

In some embodiments, user mode graphics driver 1026 contains a back-end shader compiler 1027 to convert the shader instructions 1012 into a hardware specific representation. When the OpenGL API is in use, shader instructions 1012 in the GLSL high-level language are passed to a user mode graphics driver 1026 for compilation. In some embodiments, user mode graphics driver 1026 uses operating system kernel mode functions 1028 to communicate with a kernel mode graphics driver 1029. In some embodiments, kernel mode graphics driver 1029 communicates with graphics processor 1032 to dispatch commands and instructions.

IP Core Implementations

One or more aspects of at least one embodiment may be implemented by representative code stored on a machine-readable medium which represents and/or defines logic within an integrated circuit such as a processor. For example, the machine-readable medium may include instructions which represent various logic within the processor. When read by a machine, the instructions may cause the machine to fabricate the logic to perform the techniques described herein. Such representations, known as “IP cores,” are reusable units of logic for an integrated circuit that may be stored on a tangible, machine-readable medium as a hardware model that describes the structure of the integrated circuit. The hardware model may be supplied to various customers or manufacturing facilities, which load the hardware model on fabrication machines that manufacture the integrated circuit. The integrated circuit may be fabricated such that the circuit performs operations described in association with any of the embodiments described herein.

FIG. 11 is a block diagram illustrating an IP core development system 1100 that may be used to manufacture an integrated circuit to perform operations according to an embodiment. The IP core development system 1100 may be used to generate modular, re-usable designs that can be incorporated into a larger design or used to construct an entire integrated circuit (e.g., an SOC integrated circuit). A design facility 1130 can generate a software simulation 1110 of an IP core design in a high level programming language (e.g., C/C++). The software simulation 1110 can be used to design, test, and verify the behavior of the IP core. A register transfer level (RTL) design can then be created or synthesized from the simulation model 1100. The RTL design 1115 is an abstraction of the behavior of the integrated circuit that models the flow of digital signals between hardware registers, including the associated logic performed using the modeled digital signals. In addition to an RTL design 1115, lower-level designs at the logic level or transistor level may also be created, designed, or synthesized. Thus, the particular details of the initial design and simulation may vary.

The RTL design 1115 or equivalent may be further synthesized by the design facility into a hardware model 1120, which may be in a hardware description language (HDL), or some other representation of physical design data. The HDL may be further simulated or tested to verify the IP core design. The IP core design can be stored for delivery to a 3rd party fabrication facility 1165 using non-volatile memory 1140 (e.g., hard disk, flash memory, or any non-volatile storage medium). Alternatively, the IP core design may be transmitted (e.g., via the Internet) over a wired connection 1150 or wireless connection 1160. The fabrication facility 1165 may then fabricate an integrated circuit that is based at least in part on the IP core design. The fabricated integrated circuit can be configured to perform operations in accordance with at least one embodiment described herein.

FIG. 12 is a block diagram illustrating an exemplary system on a chip integrated circuit 1200 that may be fabricated using one or more IP cores, according to an embodiment. The exemplary integrated circuit includes one or more application processors 1205 (e.g., CPUs), at least one graphics processor 1210, and may additionally include an image processor 1215 and/or a video processor 1220, any of which may be a modular IP core from the same or multiple different design facilities. The integrated circuit includes peripheral or bus logic including a USB controller 1225, UART controller 1230, an SPI/SDIO controller 1235, and an I²S/I²C controller 1240. Additionally, the integrated circuit can include a display device 1245 coupled to one or more of a high-definition multimedia interface (HDMI) controller 1250 and a mobile industry processor interface (MIPI) display interface 1255. Storage may be provided by a flash memory subsystem 1260 including flash memory and a flash memory controller. Memory interface may be provided via a memory controller 1265 for access to SDRAM or SRAM memory devices. Some integrated circuits additionally include an embedded security engine 1270.

Additionally, other logic and circuits may be included in the processor of integrated circuit 1200, including additional graphics processors/cores, peripheral interface controllers, or general purpose processor cores.

Apparatus and Method for Flexible Finite Differences in a Graphics Processor Unit

As mentioned above, GPUs typically run pixel shading on 2×2 pixel quads, packing multiple such quads into one single instruction multiple data (SIMD) batch. It often happens that one or more pixels in a quad lie outside the rasterized primitive, which are referred to as “helper” pixels. The primary need for helper pixels is for gradient computation by finite differencing, i.e., taking the difference between nearby pixels horizontally and vertically to estimate the rate at which a variable changes.

Gradients are required for the level of detail (LOD) computation in texture lookups, and also for certain shared instructions (e.g., such as ddx/ddy instructions). However, helper pixels contribute in no other way to the visible image. Since shading helper pixels consumes valuable resources (power, bandwidth, and compute), it is desirable to find less expensive ways to compute gradients. The number of helper pixels in quad-based pixel shading can be substantial, often in the range of 40-50% or more of the total shaded pixels. Additionally, the problem of helper pixels is amplified for smaller triangles, and/or with the use of larger shading quads (i.e. coarse pixel shading (CPS)).

A hierarchical rasterizer normally stamps out a tile with a large number of pixels/samples in parallel. FIG. 13 illustrates an exemplary 8×8 pixel rasterizer stamp 1302 for a triangle 1303 in which uncovered helper pixels 1301 are identified with a dotted pattern and covered pixels 1304 are identified with a solid color. In this case, the triangle 1303 covers 19 pixels and there are 17 helper pixels. Consequently, 47% of the shading is done on helper pixels resulting in an 89% overhead. This is representative of typical workloads with small/medium sized triangles.

Previous solutions to this problem include masking out SIMD lanes and/or packing pixels when gradients are not needed. This does not work, however, when the shader contains texture lookups, which is often the case. Another technique computes gradients analytically based on the plane equation of the triangle. However, this only works for first-order texture lookups (not dependent lookups) and requires additional floating-point arithmetic per pixels (in software or hardware). Yet another solution explicitly computes gradients by inserting shader code (e.g., using automatic differentiation). This solution, however, requires additional shader instructions to be executed, which is not power efficient. Supporting dependent texture lookups also requires the possibility to sample gradients (i.e., accessing individual filter taps).

The embodiments of the invention described below implement techniques for reducing the number of uncovered helper pixels by generalizing the way in which finite differences are computed. These techniques do not restrict the current usage model—i.e., dependent texture lookups are supported, the ddx/ddy instructions can compute gradients of any shader variable, and no changes are required to user assets.

FIG. 14 illustrates a pixel shader 1400 in accordance with one embodiment of the invention which includes helper pixel selection logic 1410 for selecting helper pixels using the techniques described herein. In particular, using covered and uncovered pixels 1405 as input (e.g., within an 8×8 pixel rasterizer stamp such as illustrated in FIG. 13), the helper pixel selection logic 1410 attempts to reduce the number of helper pixels which are uncovered (i.e., not part of the triangle) by generalizing the manner in which finite differences are computed and selecting “helper” pixels which are covered (i.e., pixels which are part of the triangle). Only in cases where suitable covered helper pixels cannot be identified does the helper pixel selection logic 1410 select uncovered helper pixels. The end result is that the selected set of “helper” pixels include significantly more covered pixels, thereby significantly reducing the shading operations which would otherwise be performed on uncovered “helper” pixels. The selected helper pixels are then provided to a shader module 1412 which performs a set of shading operations to generate a set of shaded pixels 1415 (using fewer uncovered helper pixels).

In one embodiment, the pixel shader 1400 is generalized to operate on individual pixels packed tightly to fill the desired SIMD width. For each covered pixel in a rasterizer stamp 1405, the helper pixel selection logic 1410 looks for two other covered pixels in its immediate vicinity (e.g. within a 3×3 pixel window). If two non-parallel axes can be found, these two pixels are used for computing gradients by way of finite differences followed by a simple transform.

In one embodiment, the computed gradients are transformed into axis-aligned screen-space gradients (as expected by the texture sampler and APIs). By limiting the search window, this transform can be implemented using a small number of very simple operations, e.g., a single addition and/or divide-by-two (shift-right of floating point exponent) per gradient.

In one embodiment, the helper pixel selection logic 1410 looks in the 3×3 neighborhood (clipped to the rasterizer stamp). FIG. 15A illustrates how horizontal differencing (ddx) may be performed and FIG. 15B illustrates how vertical differencing (ddy) may be performed. For horizontal differencing (ddx), a covered pixel to the left/right or diagonally one pixel up/down is used. Similarly, for vertical differencing (ddy), a covered pixel vertically/diagonally above or below is used. In one embodiment, if no covered helper pixels can be found, traditional, uncovered helper pixel(s) are inserted.

Current GPUs use 2×2 pixel quads, which results in a large number of helper pixels. FIG. 16 illustrates a typical example for a triangle 1603 within a rasterizer stamp 1602 with a plurality of covered pixels 1604 where the gradients for a few select pixels are marked with arrows. As illustrated, there are many more uncovered helper pixels 1601 (outside the triangle) than covered helper pixels 1605 for the select pixels marked with arrows. In total, there are nearly as many uncovered helper pixels 1601 as there are covered pixels 1604.

As illustrated in FIG. 17, in one embodiment of the invention, the same set of gradients are instead evaluated for the rasterizer stamp 1602 using covered helper pixels 1701 which are all part of the triangle. Hence, in this case, no uncovered helper pixels need to be introduced, saving 17 unnecessary shader evaluations (47% of the total in this example). Note that this example assumes the covered pixels can all fit in one SIMD batch.

For narrower SIMD widths, the helper pixel selection logic 1410 attempts to fill each SIMD batch with pixels that are not already shaded and that contribute to the computation of gradients. Only if these conditions are not met, are traditional helper pixel(s) inserted.

One embodiment of the invention allows other pixels than those inside the current 2×2 quad to be used for computing gradients by finite differencing. It performs these operations by selecting pixels that are already covered by the rasterized primitive (to the extent possible). Within the current SIMD batch, shading of useless extra helper pixels can thereby be largely avoided.

A method for selecting helper pixels in accordance with one embodiment of the invention is illustrated in FIG. 18. The method may be implemented within the context of the architectures described above, but is not limited to any particular graphics processor architectures.

At 1801 the first/next pixel within a rasterizer stamp is selected and, at 1802, a determination is made as to whether a suitable covered pixel exists which may be used as a helper pixel for horizontal differencing. If so, then at 1803, the covered pixel is selected for horizontal differencing (e.g., using the techniques described herein to selected the most suitable pixel). If not, then at 1804, an uncovered pixel is selected for horizontal differencing.

At 1805, a determination is made as to whether a suitable covered pixel exists for vertical differencing. If so, then at 1807, the covered pixel is selected for vertical differencing. If not, then at 1806, an uncovered pixel is selected for vertical differencing. If additional pixels remain within the rasterizer stamp which require helper pixels, determined at 1808, then the process returns to 1801 and the next pixel is selected. If not, then the process ends for the current rasterizer stamp.

Note that the sequential nature of the flowchart illustrated in FIG. 18 is used for the purposes of illustration but is not intended to indicate that the operations described therein must be performed in a serial manner. For example, in one embodiment, the illustrated operations may be performed in parallel within a graphics processor capable of simultaneous multithreading and/or SIMD operations.

The following additional details are provided for one strategy designed for a specific hardware implementation. It should be noted, however, that the underlying principles of the invention are not limited to these specific details. Many other embodiments are possible using various different implementation details (e.g., a larger search window, different rules for pixel selection, etc).

Flexible Finite Differencing

Looking at pixel with screen coordinates x=(x, y), two helper pixels are chosen at x+v where u=(u_(x), u_(y)) and v=(v_(x), v_(y)) for approximating the gradient of an attribute f(.) through finite differences. The gradients are denoted Δ_(u)=f(x+u)−f(x) and similar for Δ_(v).

For compatibility with the texture sampler and current APIs (e.g., ddx/ddy instructions), the gradients are then transformed to screen-space gradients. For example: Δ_(x)=f(x+1)−f(x), using a 2D transform. Note that the u and v axes must be chosen to be non-parallel.

The transform to screen-space is given by:

$\begin{matrix} {\begin{pmatrix} x \\ y \end{pmatrix} = {\left. {A\begin{pmatrix} u \\ v \end{pmatrix}}\leftrightarrow\begin{pmatrix} u \\ v \end{pmatrix} \right. = {{{A^{- 1}\begin{pmatrix} x \\ y \end{pmatrix}}\mspace{14mu} {where}\mspace{14mu} A} = {\begin{pmatrix} u_{x} & v_{x} \\ u_{y} & v_{y} \end{pmatrix}.}}}} & \left( {{Eqn}\mspace{11mu} 1} \right) \end{matrix}$

The inverse matrix is:

$A^{- 1} = {{\frac{1}{\det \mspace{11mu} A}\begin{pmatrix} v_{y} & {- v_{x}} \\ {- u_{y}} & u_{x} \end{pmatrix}} = {\frac{1}{{u_{x}v_{y}} - {v_{x}u_{y}}}{\begin{pmatrix} v_{y} & {- v_{x}} \\ {- u_{y}} & u_{x} \end{pmatrix}.}}}$

The right hand side of Eqn 1 indicates how to weigh the two gradients, Δ_(u) and Δ_(v), to get the horizontal/vertical gradients by inserting (x, y)=(1, 0) and (0, 1) respectively.

Simplifications

Looking at the 3×3 neighborhood around a pixel, it is noted that u_(x), u_(y), v_(x), v_(y)ε{−1, 0, 1}, and det A≠0 (since the two axes are non-parallel). Hence, the value of 1/detA can only take four possible values: ±1, ±0.5.

Using the rules outlined below there are 36 possible combinations of helper pixel locations by selecting the first as any of the 6 pixels left/right of the current pixel, and the other as any of 6 positions above/below (note that some of these combinations are invalid). Hence, the computation of 1/detA can be implemented in a very compact fixed-function lookup table (gate network) with a 36-to-4 mapping or smaller. The output can be described as a two-bit value that selects: 1) sign, 2) divide-by-two (i.e. shift-right of floating point exponent by one bit). Under these constraints (i.e., 3×3 region), transforming the gradients to screen-space thus reduces to possible sign-flips, a single addition, and possibly shift-right per gradient. All these operations can be efficiently implemented in hardware or software.

For example, if u=(1,1) and v=(0,−1), the forward horizontal/vertical gradients are sought (evaluated at x+1 and y+1), and the result is:

$\Delta_{x} = {{\frac{1}{- 1}\begin{pmatrix} {- 1} & {- 1} \end{pmatrix}\begin{pmatrix} \Delta_{u} \\ \Delta_{v} \end{pmatrix}} = {{\Delta_{u} + {\Delta_{v}\mspace{14mu} {and}\mspace{14mu} \Delta_{y}}} = {- {\Delta_{v}.}}}}$

This may be simple to verify geometrically by looking at the two vectors.

Rules for Selecting Helper Pixels

In the traditional GPU pipeline, finite differences are evaluated horizontally/vertically within each 2×2 pixel quad in the rasterizer stamp. Hence, u=(±1,0) and v=(0, ±1) where positive signs are used on even pixels (top/left) and negative on odd pixels (bottom/right) of the quad.

In one embodiment of the invention, the same principles are followed as far as possible. For example, in selecting the helper pixel for the horizontal gradient, a search right is performed for even pixels and a search left for odd pixels. If no covered pixels are available in that direction, or if the location is at the edge of the rasterizer stamp, a search is performed in the other direction. The same is done for vertical gradients, but looking up or down. These rules ensure that in the interiors of primitives, finite differences are evaluated exactly the same way as in current GPUs.

If no suitable covered pixels can be found in either direction, a traditional helper pixel is inserted. This is necessary, for example, for small primitives and sliver triangles, to ensure that two helper pixels with non-parallel axes are available for each covered pixel. However, the number of such inserted helper pixels is much smaller than enforcing conventional 2×2 quads. By way of example, FIG. 19A illustrates traditional 2×2 quads 1901 resulting in substantial over-shading for small primitives due to a large number of uncovered helper pixels. For example, uncovered helper pixel 1902 is selected for triangle 1903.

In contrast, using the embodiments of the invention as shown in FIG. 19B, the number of helper pixels is significantly reduced (to zero in some cases). Specifically, a covered helper pixel 1912 is selected within triangle 1903 rather than the uncovered helper pixel 1902 shown in FIG. 19B. Similarly, each of the two covered pixels within triangle 1910 selects the other as a helper pixel, thereby reducing the number of helper pixels from four (in FIG. 19A) to two (in FIG. 19B) for the triangle 1910.

In one embodiment, when the number of suitable covered pixels in the 3×3 neighborhood is insufficient around each pixel, additional uncovered helper pixels may be inserted. For example, as shown in FIG. 20, the number of covered pixels for triangle 2001 is insufficient and therefore two uncovered helper pixels 2002 are selected. Note that the topmost helper pixel can here be used by the two topmost covered pixels, but at the third pixel down, a suitable helper to the left/right is missing. For the remaining pixels, it is always possible to select two helper pixels that are already covered.

In one embodiment, shading is evaluated by the shader cores in a SIMD fashion. For example, 8, 16 or 32 pixels may be shaded together in one SIMD batch. To maximize utilization, one embodiment of the invention fits as many pixels with valid neighbors for gradient computation in each SIMD batch.

One embodiment traverses the rasterizer stamp in Morton (Z) order and incrementally adds covered pixels that are not already shaded to the current SIMD batch. As is known by those of skill in the art, “Z-order” or “Morton order” is a function which maps multidimensional data to one dimension while preserving locality of the data points. For each such pixel added, neighboring pixels are searched for in the 3×3 window around the current pixel. In one embodiment, the selection of neighbors is prioritized according to:

1. Pixels that are already included in the current SIMD batch (i.e., since these pixels do not incur any extra cost).

2. Pixels that are covered and not shaded in a previous SIMD batch (i.e., since these pixels represent useful work).

3. Other pixels, such as those uncovered pixels used as traditional helper pixels.

Once a SIMD batch is full or no more pixels can be added, one embodiment checks for each pixel in the batch whether it has two valid neighboring pixels shaded in the same batch. If so, the pixel is marked as “valid” and its result is used not only as a helper pixel. This may enable additional pixels that were added under (2) above.

Quality Assessment

For attributes that are linear in screen space, the embodiments of the invention described herein will yield exact gradients (i.e., the same as finite differencing on 2×2 quads). For non-linear attributes, the results will potentially differ for pixels along the edges of primitives (interiors evaluate the same). This is a similar tradeoff to what centroid sampling introduces. However, in that case, the gradients are not corrected for the placement of the centroid, causing severe artifacts. In the instant case, any differences are due to the non-linearity of the evaluated attribute.

For dependent texture lookups, the embodiments of the invention may generate superior results than traditional finite differences since the goal is to choose helper pixels that lie inside the primitive, and thus have valid texture values. Using 2×2 quads, on the other hand, the evaluated helper pixels often lie outside the primitive, and may be sampling undefined regions of texture atlases, etc.

Other Considerations

Since the gradient computation may affect image quality (compared to shading on 2×2 quads), one embodiment provides the user control through APIs and/or control panel settings. This is similar to how users currently can choose between center-of-pixel vs centroid sampling.

While the embodiments of the invention have been described in the context of pixel shading, they are not limited to this application. For example, with coarse pixel shading using larger-than-pixel-sized shading quads, gradients over such quads may be evaluated using the techniques described herein. For object-space shading, for example, using an arbitrary shading space with a shading memoization cache or deferred solution, one embodiment of the invention may be used if shading requests are buffered up and not speculatively shaded (over 2×2 quads) on a first miss.

While the embodiments of the invention have been described in the context of pixel shading with a single sample per pixel, they are not limited to this application. For example, multiple samples per pixel may be used.

In contrast to previous solutions such as not computing gradients when it is safe to do so (e.g., no LOD computations or ddx/ddy instructions) and explicitly computing gradients using analytical or automatic differentiation the embodiments of the invention work generally for all cases and may be implemented using existing operations (e.g., swizzle, sign flip, addition, and shift-right). This avoids most of the software overhead, at the cost of potentially shading a small set of helper pixels.

The chart in FIG. 21 shows the end-to-end reduction in the number of pixel shader invocations over an entire frame of a particular video game (“Civilization 5”). Note that only about 45% of the total work is small/medium triangles where our invention helps. The configuration used for this test is a conceptual GPU with 4 graphics cores, 8×8 pixel rasterizer stamps, and a resolution 1920×1200 pixels.

In this detailed description, reference is made to the accompanying drawings, which form a part hereof, wherein like numerals designate like parts throughout, and in which is shown by way of illustration embodiments that may be practiced. It is to be understood that other embodiments may be utilized and structural or logical changes may be made without departing from the scope of the present disclosure. Therefore, the following detailed description is not to be taken in a limiting sense, and the scope of embodiments is defined by the appended claims and their equivalents.

Various operations may be described as multiple discrete actions or operations in turn, in a manner that is most helpful in understanding the claimed subject matter. However, the order of description should not be construed as to imply that these operations are necessarily order dependent. In particular, these operations may not be performed in the order of presentation. Operations described may be performed in a different order than the described embodiment. Various additional operations may be performed and/or described operations may be omitted in additional embodiments.

For the purposes of the present disclosure, the phrase “A and/or B” means (A), (B), or (A and B). For the purposes of the present disclosure, the phrase “A, B, and/or C” means (A), (B), (C), (A and B), (A and C), (B and C), or (A, B, and C). Where the disclosure recites “a” or “a first” element or the equivalent thereof, such disclosure includes one or more such elements, neither requiring nor excluding two or more such elements. Further, ordinal indicators (e.g., first, second, or third) for identified elements are used to distinguish between the elements, and do not indicate or imply a required or limited number of such elements, nor do they indicate a particular position or order of such elements unless otherwise specifically stated.

Reference in the description to one embodiment or an embodiment means that a particular feature, structure, or characteristic described in connection with the embodiment is included in at least one embodiment of the invention. The description may use the phrases “in one embodiment,” “in another embodiment,” “in some embodiments,” “in embodiments,” “in various embodiments,” or the like, which may each refer to one or more of the same or different embodiments. Furthermore, the terms “comprising,” “including,” “having,” and the like, as used with respect to embodiments of the present disclosure, are synonymous.

In embodiments, the term “engine” or “module” or “logic” may refer to, be part of, or include an application specific integrated circuit (ASIC), an electronic circuit, a processor (shared, dedicated, or group), and/or memory (shared, dedicated, or group) that execute one or more software or firmware programs, a combinational logic circuit, and/or other suitable components that provide the described functionality. In embodiments, an engine or a module may be implemented in firmware, hardware, software, or any combination of firmware, hardware, and software.

Embodiments of the invention may include various steps, which have been described above. The steps may be embodied in machine-executable instructions which may be used to cause a general-purpose or special-purpose processor to perform the steps. Alternatively, these steps may be performed by specific hardware components that contain hardwired logic for performing the steps, or by any combination of programmed computer components and custom hardware components.

As described herein, instructions may refer to specific configurations of hardware such as application specific integrated circuits (ASICs) configured to perform certain operations or having a predetermined functionality or software instructions stored in memory embodied in a non-transitory computer readable medium. Thus, the techniques shown in the figures can be implemented using code and data stored and executed on one or more electronic devices (e.g., an end station, a network element, etc.). Such electronic devices store and communicate (internally and/or with other electronic devices over a network) code and data using computer machine-readable media, such as non-transitory computer machine-readable storage media (e.g., magnetic disks; optical disks; random access memory; read only memory; flash memory devices; phase-change memory) and transitory computer machine-readable communication media (e.g., electrical, optical, acoustical or other form of propagated signals—such as carrier waves, infrared signals, digital signals, etc.).

In addition, such electronic devices typically include a set of one or more processors coupled to one or more other components, such as one or more storage devices (non-transitory machine-readable storage media), user input/output devices (e.g., a keyboard, a touchscreen, and/or a display), and network connections. The coupling of the set of processors and other components is typically through one or more busses and bridges (also termed as bus controllers). The storage device and signals carrying the network traffic respectively represent one or more machine-readable storage media and machine-readable communication media. Thus, the storage device of a given electronic device typically stores code and/or data for execution on the set of one or more processors of that electronic device. Of course, one or more parts of an embodiment of the invention may be implemented using different combinations of software, firmware, and/or hardware. Throughout this detailed description, for the purposes of explanation, numerous specific details were set forth in order to provide a thorough understanding of the present invention. It will be apparent, however, to one skilled in the art that the invention may be practiced without some of these specific details. In certain instances, well known structures and functions were not described in elaborate detail in order to avoid obscuring the subject matter of the present invention. Accordingly, the scope and spirit of the invention should be judged in terms of the claims which follow. 

What is claimed is:
 1. A graphics processor comprising: pixel shading logic to perform pixel shading operations on pixels associated with a rasterized primitive using covered pixels and uncovered pixels; and helper pixel selection logic to select helper pixels in the rasterized primitive, the helper pixels to be used by the pixel shading logic for gradient computations, wherein for one or more of the covered pixels, the helper pixel selection logic attempts to identify one or more suitable covered helper pixels and, if no suitable covered helper pixels exist, identifies one or more uncovered helper pixels.
 2. The graphics processor as in claim 1 wherein for each covered pixel for which a helper pixel is selected, the helper pixel selection logic is to select a first helper pixel for horizontal differencing and a second helper pixel for vertical differencing.
 3. The graphics processor as in claim 2 wherein horizontal differencing is performed by searching for a covered helper pixel to the left, right or diagonally one pixel up or down from each covered pixel for which a helper pixel is selected.
 4. The graphics processor as in claim 3 wherein vertical differencing is performed by searching for a covered helper pixel vertically above or below, or diagonally above or below each covered pixel for which a helper pixel is selected.
 5. The graphics processor as in claim 2 wherein if a suitable covered helper pixel cannot be found for vertical differencing, an uncovered helper pixel is selected for vertical differencing, and if a suitable covered pixel cannot be found for horizontal differencing, an uncovered helper pixel is selected for horizontal differencing.
 6. The graphics processor as in claim 1 wherein covered pixels comprise pixels which are covered by the rasterized primitive and uncovered pixels comprise pixels which are not covered by the rasterized primitive.
 7. The graphics processor as in claim 1 wherein the pixel shading logic is to perform gradient computations to determine gradients by finite differencing.
 8. The graphics processor as in claim 7 wherein the pixel shading logic is to transform the gradients to screen-space gradients.
 9. The graphics processor as in claim 8 wherein the pixel shading logic transforms the gradients to screen-space gradients according to a first equation: $\begin{matrix} {\begin{pmatrix} x \\ y \end{pmatrix} = {\left. {A\begin{pmatrix} u \\ v \end{pmatrix}}\leftrightarrow\begin{pmatrix} u \\ v \end{pmatrix} \right. = {{{A^{- 1}\begin{pmatrix} x \\ y \end{pmatrix}}\mspace{14mu} {where}\mspace{14mu} A} = {\begin{pmatrix} u_{x} & v_{x} \\ u_{y} & v_{y} \end{pmatrix}.}}}} & \left( {{Eqn}\mspace{11mu} 1} \right) \end{matrix}$ where the inverse matrix is defined in: $A^{- 1} = {{\frac{1}{\det \mspace{11mu} A}\begin{pmatrix} v_{y} & {- v_{x}} \\ {- u_{y}} & u_{x} \end{pmatrix}} = {\frac{1}{{u_{x}v_{y}} - {v_{x}u_{y}}}{\begin{pmatrix} v_{y} & {- v_{x}} \\ {- u_{y}} & u_{x} \end{pmatrix}.}}}$ and where the right hand side of the first equation indicates how to weigh the two gradients, Δ_(u) and Δ_(v), to get the horizontal and vertical gradients by inserting (x, y)=(1, 0) and (0, 1) respectively.
 10. The graphics processor as in claim 9 wherein the computation of 1/detA is implemented in a fixed-function lookup table with a 36-to-4 mapping or smaller.
 11. A machine readable medium having program code stored thereon which, when executed by a graphics processor, causes the graphics processor to perform the operations of: performing pixel shading operations on pixels associated with a rasterized primitive using covered pixels and uncovered pixels; and selecting helper pixels in the rasterized primitive, the helper pixels to be used for gradient computations when performing the pixel shading operations, attempting to identify one or more suitable covered helper pixels for performing the gradient computations for one or more of the covered pixels and, if no suitable covered helper pixels exist, identifying one or more uncovered helper pixels.
 12. The machine-readable medium as in claim 11 wherein for each covered pixel for which a helper pixel is selected, the program code executed by the graphics processor selects a first helper pixel for horizontal differencing and a second helper pixel for vertical differencing.
 13. The machine-readable medium as in claim 12 wherein horizontal differencing is performed by searching for a covered helper pixel to the left, right or diagonally one pixel up or down from each covered pixel for which a helper pixel is selected.
 14. The machine-readable medium as in claim 13 wherein vertical differencing is performed by searching for a covered helper pixel vertically above or below, or diagonally above or below each covered pixel for which a helper pixel is selected.
 15. The machine-readable medium as in claim 12 wherein if a suitable covered helper pixel cannot be found for vertical differencing, an uncovered helper pixel is selected for vertical differencing, and if a suitable covered pixel cannot be found for horizontal differencing, an uncovered helper pixel is selected for horizontal differencing.
 16. The machine-readable medium as in claim 11 wherein covered pixels comprise pixels which are covered by the rasterized primitive and uncovered pixels comprise pixels which are not covered by the rasterized primitive.
 17. The machine-readable medium as in claim 11 wherein the program code causes the graphics processor to perform gradient computations to determine gradients by finite differencing.
 18. The machine-readable medium as in claim 17 wherein the program code further causes the graphics processor to transform the gradients to screen-space gradients.
 19. The machine-readable medium as in claim 18 wherein the program code is to further cause the graphics processor to transform the screen space gradients according to a first equation: $\begin{matrix} {\begin{pmatrix} x \\ y \end{pmatrix} = {\left. {A\begin{pmatrix} u \\ v \end{pmatrix}}\leftrightarrow\begin{pmatrix} u \\ v \end{pmatrix} \right. = {{{A^{- 1}\begin{pmatrix} x \\ y \end{pmatrix}}\mspace{14mu} {where}\mspace{14mu} A} = {\begin{pmatrix} u_{x} & v_{x} \\ u_{y} & v_{y} \end{pmatrix}.}}}} & \left( {{Eqn}\mspace{11mu} 1} \right) \end{matrix}$ where the inverse matrix is defined in: $A^{- 1} = {{\frac{1}{\det \mspace{11mu} A}\begin{pmatrix} v_{y} & {- v_{x}} \\ {- u_{y}} & u_{x} \end{pmatrix}} = {\frac{1}{{u_{x}v_{y}} - {v_{x}u_{y}}}{\begin{pmatrix} v_{y} & {- v_{x}} \\ {- u_{y}} & u_{x} \end{pmatrix}.}}}$ and where the right hand side of the first equation indicates how to weigh the two gradients, Δ_(u) and Δ_(v), to get the horizontal and vertical gradients by inserting (x, y)=(1, 0) and (0, 1) respectively.
 20. The machine-readable medium as in claim 19 wherein the computation of 1/detA is implemented in a fixed-function lookup table with a 36-to-4 mapping or smaller. 